Form Factors

The physical landscape may be represented by a function $I(\ensuremath{{\mathbf{r}}}\xspace )$ describing a potential-determining property such as the local optical intensity. An object's potential energy at $\ensuremath{{\mathbf{r}}}\xspace$ is determined, not only by $I(\ensuremath{{\mathbf{r}}}\xspace )$, but also by the object's response to it. For example, larger particles approaching a well-localized optical trap encounter the trap's intensity gradients at larger ranges than smaller particles. The observation that different objects passing through the same environment experience different potential-energy landscapes provides the foundation for the results that follow.

The effective potential may be expressed as the convolution

$$V(\ensuremath{{\mathbf{r}}}\xspace ) = (f \circ I)(\ensuremath{{\mathbf{r}}}\xspace )$$ (2)

$$= \int f(\vec{x} - \ensuremath{{\mathbf{r}}}\xspace ) \, I(\vec{x}) \, d^2x,$$ (3)

of the two-dimensional landscape, $I(\ensuremath{{\mathbf{r}}}\xspace )$, with a form factor $f(\ensuremath{{\mathbf{r}}}\xspace )$ describing the object's interaction with the landscape. In comparing to experimental realizations, we assume that contributions from the form factor's third dimension have been integrated out. If $I(\ensuremath{{\mathbf{r}}}\xspace )$ has a symmetry axis along the $\hat x$ direction, then the associated force,

$$\vec{F}(\ensuremath{{\mathbf{r}}}\xspace ) = - \vec{\nabla} \, (f \circ I)(\ensuremath{{\mathbf{r}}}\xspace ),$$ (4)

generally does as well. Convolving with $f(\ensuremath{{\mathbf{r}}}\xspace )$ broadens features in $I(\ensuremath{{\mathbf{r}}}\xspace )$ by an amount that depends on the object's size, shape, orientation, and composition.

In many cases of practical interest, the convolution in Eq. (2) is most easily performed using the Fourier convolution theorem:

$$(f \circ I)(\ensuremath{{\mathbf{r}}}\xspace ) = {\cal F}^{-1} \{\tilde f(\vec{k}) \, \tilde I(\vec{k})\}$$ (5)

where $\tilde f(\vec{k})$ and $\tilde I(\vec{k})$ are the Fourier transforms of $f(\ensuremath{{\mathbf{r}}}\xspace )$ and $I(\ensuremath{{\mathbf{r}}}\xspace )$, respectively, and $g(\ensuremath{{\mathbf{r}}}\xspace ) = {\cal F}^{-1}\{\tilde g(\vec{k})\}$ denotes the inverse Fourier transform of $\tilde g(\vec{k})$. In some particularly simple cases, both $\tilde f(\vec{k})$ and $\tilde I(\vec{k})$ can be factored into components along the $\hat x$ and $\hat y$ directions, reducing Eq. (5) to a product of one-dimensional integrals. In other cases, separable approximations for the form factor emerge as the leading-order cumulant expansion of $\tilde f(\vec{k})$.

For example, the form factor for a uniform dielectric cube of side $a$ aligned with the $x$ axis and illuminated by collimated light is

$$f(\ensuremath{{\mathbf{r}}}\xspace ) = \alpha a \, \Theta \left( ... ...a}{2}-\lvert x \rvert \right) \Theta \left( \frac{a}{2}-\lvert y \rvert \right)$$ (6)

where $\Theta(x)$ is the Heaviside step function, and

$$\alpha = 2 \pi \, \frac{\sqrt{\epsilon_0}}{c} \, \left( \frac{\epsilon_0 - \epsilon}{\epsilon + 2 \epsilon_0}\right)$$ (7)

describes the matter-light interaction, in the quasi-static limit, for a material of dielectric constant $\epsilon$ immersed in a medium of dielectric constant $\epsilon_0$ (25). Both this geometry and the collimated light field are far simpler than would be encountered in most real-world optical trapping implementations (26), but serve to illustrate our approach. A more complete treatment of optical forces also would incorporate polarization effects, which cannot be captured in the present scalar theory. Higher-order effects such as Mie resonances (25) could be taken into account through $\alpha$, but will be ignored in the current discussion. Note that $\alpha$ is negative for a high-dielectric-constant material in a low-dielectric-constant medium; such particles are drawn toward regions of high intensity. Low-dielectric-constant particles, by contrast, are repelled by light.

The aligned cube's form factor is separable, with Fourier transform

$$\tilde f(\vec{k}) = \alpha \, a^3 \, \tilde f_x(k_x a) \, \tilde f_y(k_y a).$$ (8)

The individual components are readily shown to be

$$\tilde f_x(ka) = \tilde f_y(ka) = \frac{\sin ka}{ka}.$$ (9)

Their leading-order cumulant expansion,

$$\tilde f_x(ka) = \tilde f_y(ka) \approx \exp\left(- \frac{1}{6} \, k^2 a^2\right)$$ (10)

for $ka < \pi$, demonstrates that the form factor's Fourier transform depends sensitively on particle size for a given wavenumber. Note that, as defined, $\tilde f_x(ka)$ and $\tilde f_y(ka)$ are dimensionless and normalized to unity at $ka = 0$.

The form factor for a uniform dielectric sphere of radius $a$ illuminated by collimated light of wavelength $\lambda > a$ is (25),

$$f(\ensuremath{{\mathbf{r}}}\xspace ) = \alpha \, \sqrt{a^2 - r^2} \, \Theta(a-r),$$ (11)

which is not separable. The leading-order cumulant expansion of $\tilde f(\vec{k})$, however, is separable, with

$$\tilde f(\vec{k}) \approx \alpha \, \frac{2 \pi a^3}{3} \, \tilde f_x(k_x a) \, \tilde f_y(k_y a) \, ,$$ (12)

where

$$\tilde f_x(ka) = \tilde f_y(ka) = \exp\left(-\frac{1}{10} \, k^2a^2\right),$$ (13)

for $k a < 8$.

More generally, an object's form factor is nonzero only over a limited domain, set by its size. The corresponding Fourier transform thus depends strongly on $ka$, within the appropriate range of wavenumbers. We capture the ramifications of this boundedness by adopting the separable Gaussian form

$$f(\ensuremath{{\mathbf{r}}}\xspace ) = \alpha \, a \, \exp\left( - \frac{r^2}{2 a^2} \right),$$ (14)

whose Fourier transform

$$\tilde f (\vec{k}) = 2\pi \alpha a^3 \, \tilde f_x(k_x a) \, \tilde f_y(k_y a)$$ (15)

has components

$$\tilde f_x(ka) = \tilde f_y(ka) = \exp\left( - \frac{1}{2} k^2 a^2 \right).$$ (16)

Which wavenumbers come into play depends on the landscape, $I(\ensuremath{{\mathbf{r}}}\xspace )$. The following Sections explore a few particularly effective choices.